some comments on gravitational entropy and the inverse mean curvature flow

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Some comments on gravitational entropy and the inverse mean curvature flow

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Class. Quantum Grav. 16 (1999) 16771687. Printed in the UK PII: S0264-9381(99)98512-X

Some comments on gravitational entropy and the inverse

mean curvature flow

G W Gibbons

DAMTP, Cambridge University, Silver Street, Cambridge CB3 9EW, UK

Received 15 October 1998

Abstract. The GerochWaldJangHuiskenIlmanen approach to the positive energy problemmay be extended to give a negative lower bound for the mass of asymptotically anti-de Sitterspacetimes containing horizons with exotic topologies having endsor infinitiesof the form g

R,

in terms of the cosmological constant. We also show how the method gives a lower bound for themass of time-symmetric initial data sets for black holes with vectors and scalars in terms of themass, |Z(Q,P)| of thedouble-extreme black hole with thesame charges. I also give a lower boundfor the area of an apparent horizon, and hence a lower bound for the entropy in terms of the samefunction |Z(Q,P)|. This shows that the so-called attractor behaviour extends beyond the staticspherically symmetric case. and underscores the general importance of the function |Z(Q,P)|.There are hints that higher-dimensional generalizations may involve the Yamabe conjectures.

PACS numbers: 0240, 0420, 0470

1. Introduction

Recently, Huisken and Ilmanen [4, 5] have made mathematically rigorous an old idea of

Gerochs [1] for proving the positive mass theorem using the inverse mean curvature flow.

It was realized by Jang and Wald [2] soon after Gerochs suggestion that if the method could

be made to work it would yield a lower bound for the mass of an asymptotically flat spacetime

in terms of the area of any apparent horizon. The inverse mean curvature flow is not in general

smooth and has jumps. Nevertheless Huisken and Ilmanen are able to show that despite the

jumps the basic idea of Geroch goes through because the functional that he introduced is

monotonic through a jump.

Here I want to point out that Gerochs formal argument may be extended to cover exotic

black holes with non-trivial topology which occur in theories with a negative cosmological

term and to cover black holes with scalar and Abelian vector fields; that is locally the flow

will be monotonic. It seems reasonable to believe that the rigorous methods of Huisken and

Ilmanen can then be extended to this setting to give a genuine proof. Some of the results below

have been known to me for many years but in the absence of a proper proof for the existence ofthe Geroch flow I hesitated to publish them. In addition they were of limited cosmological or

astrophysical interest. Publication seems more appropriate now that a proper proof is at hand

and in the light of the prominent role that these black holes play in current attempts to derive

black hole entropy in terms of microstates and in applications to conformally invariant gauge

theories via the AdS/CFT correspondence. It is precisely rigorous bounds on the classical

entropy of initial data sets which could substantiate some of the ideas currently labelled with

the name holography

0264-9381/99/061677+11$19.50 1999 IOP Publishing Ltd 1677


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1678 G W Gibbons

Asymptotically flat black holes with scalar and vector fields carrying electric charges

Q and magnetic charges P arise in supergravity and supergravity theories. Their properties

are governed by a function V ( , Q , P ) quadratic in charges and depending on the manifold

in which the scalars take their values. We define a function Z(Q,P) which is the value of

V ( , Q , P ) at its critical point. The bounds we obtain on the entropy and mass may beexpressed entirely in terms of Z(Q,P). In supersymmetric theories Z(Q,P) is related to

the central charges of the theory but it may be defined more generally. Previous work has

largely been in the context of static spherically symmetric solutions (see [8, 9] and references

therein). The main point being made here is that V ( , Q , P ) and Z(Q,P) retain their

importance when considering time-dependent and non-spherically symmetric situations. This

is completely consistent with their microscopic interpretation in terms of states of D-branes.

Moreover, it provides a remarkable link between thermodynamic ideas and global differential

geometry which may well extend much further. Some links between entropy, complexity and

gravitational action in the context of hyperbolic geometry have already been made [12, 13].

These general thermodynamic ideas extend to all dimensions. However, at present the

inverse mean curvature flow techniques are restricted to (3+1)-dimensional physics. In the last

section of this paper I indicate the difficulties one encounters. It appears that overcoming themmay involve the Yamabe conjectures, a subject which has also been applied to gravitational

entropy in a cosmological context [14, 15]. In fact, the calculations in [14, 15] involved the

entropy of the matter in a self-consistent solution of the Einstein equations resulting in an

Einstein static universe, ES U4 R S3. Since ES U is both the universal cover of theconformal compactification of four-dimensional Minkowski spacetimeE3,1 and the conformal

boundary of five-dimensional anti-de Sitter spacetime AdS5, it is not inconceivable that the

AdS/CFT correspondence may entail the Yamabe conjectures in some fundamental way.

2. Time-symmetric initial data sets

We shall consider three-dimensional initial data sets {, gij} for general relativity with acosmological term. The arguments establishing lower bounds for the area of minimal 2-

surfaces will also go through if we merely demand that the initial data set is maximal (i.e.

if the trace K = gij Kij of the second fundamental form Kij vanishes). That is because thetrace-free part of the extrinsic curvature contributes positively to the Ricci scalar. Of course in

the non-time-symmetric case an apparent horizon need not be minimal. Nevertheless one may

argue that the area of the minimal surface should provide a lower bound for the area of any

apparent horizon, precisely because it is minimal. However, since the inclusion of this extra

contribution to the Ricci scalar is entirely straightforward and introduces no new features, I

shall not include it in what follows.

The Ricci scalar R of the initial metric is constrained to satisfy

R = 2 + 16 T00 (1)where T00 is the energy density of the matter and is the cosmological constant.

A basic model example is given by

ds2 = dr2 + r 2 d2k (2)where d2k is the metric of constant Gaussian curvature k = 0, 1 and

= k 2Mr

3

r2. (3)

This 3-metric is just that induced on the constant time hypersurfaces of the Kottler or

Schwarzschildde Sitter static spacetime

ds2 = dt2 + gij dxi dxj . (4)


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Gravitational entropy and the inverse mean curvature flow 1679

We are mainlyinterested in thecaseswhen the 2-surfaces of constant r are closed and orientable

with genusg. Thusifk = 1 we have 2-sphereswith g = 0,ifk = 0 we havetoriwithg = 1andifk = 1 we have g 2. We may view this last case as H2/ where H2 is two-dimensionalhyperbolic space and is a suitable discrete subgroup of its isometry group S(2, 1).

The interesting new cases occur when < 0 [7]. We then let = 3/a2 and thus

= r2

a2+ k 2M

r. (5)

IfM = 0 we have anti-de Sitter spacetime, AdS4, or a quotient of it by a discrete group. We may think ofAdS4 as a quadric in E

3,2 :

(X0)2 + (X4)2 (X1)2 (X2)2 (X3)2 = 3

. (6)

The isometry group of AdS4 is SO (3, 2). The three (locally) static forms of the metric

correspond to the three different types of one-parameter subgroups of SO (2, 1) SO (3, 2)acting say on the coordinates (X0, X4, X3).

Thus the globally static case k

=1 corresponds to rotations in the X0, X4 2-plane. The

case k = 1 corresponds to boosts in the X0, X3, and the discrete group SO (2, 1) actingin the X4, X1, X2 3-plane. We have a Killing horizon and with respect to this Killing field

AdS4 has temperature1

2

1

3.

The case k = 0 corresponds to null rotations and there is a degenerate Killing horizon Ifone identifies the horizon to obtain a torus the identifications are also null rotations. However,

these identifications do not act freely on the horizon and introduce orbifold singularities there.

Now if we pass to the Kottler solution when M = 0 we find that if k 0 then M > 0guarantees the existence of a regular apparent horizon of area A = 4 r 2H, where rH is theoutermost root of(rH) = 0. In fact,

2M = r3H

a2+ krH. (7)

Ifk = 0 and M = 0 then the spatial section is complete and has a cusp at r = 0 which is aninfinite spatial distance. The static Kottler spacetime has a degenerate Killing horizon at thecusp in this case. Ifk = 1 we have

2M = r3H

a2+ krH. (8)

Thus M can be negative but not too negative. If

M >1

27a2(9)

then we have a regular apparent horizon of area A = 4 |1 g|r 2H. The case of the equality

M = 127a2

(10)

gives a cusp at

r = 13a2

. (11)

These observations suggest a general lower bound, valid for all values of k of the form

2M1

a2

A

4

3/2+ k

A

4

1/2, (12)


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1680 G W Gibbons

where A is the area of outermost minimal surface. We shall shortly argue that precisely this

lower bound is obtainable form the inverse mean curvature flow. In particular, this seems to

implythatthe negative masstopologically non-trivial black hole is classically stable. Moreover,

the degenerate zero temperature limiting solution should be quantum mechanically stable even

thoughit is nota BPSstate, that is, it hasno Killing spinors andhenceadmits no supersymmetry.

3. The Geroch flow

The argument initiated by Geroch [1], extended by Jang and Wald [2] and completed by

Huisken and Ilmanen [4, 5] goes roughly as follows. One considers a family of level sets Sswhich are smoothly immersed 2-surfaces each with metric hab , second fundamental form pabof area A(s) mean curvature p = hab pab and Gaussian curvature KG evolving in s accordingto the inverse mean curvature flow equation so the velocity vn of the surface Ss along its normal

n is given by

vn =1

p. (13)

It follows that

dA

ds= A. (14)

One associates with each surface the function

f(s) =

Ss

(4KG p2 43 ) dA (15)

and finds that

d(f A1/2)

ds A1/2

Ss

16T00. (16)

Equality is possible if andonly if thetrace-free part of theextrinsic curvature of thesurfaces

S(s) vanishes and p is constant on the surfaces. Now start the flow fromthe outermost apparent

horizon (which of course may not be connected) on which p = 0. One assumes that the flowreaches infinity near which the metric behaves like the model example. If the surfaces tending

to r = constant surfaces, one finds thatlim f A1/2 = 64 3/2M. (17)

If the matterenergy density T00 is non-negative one may integrate the inequality to obtain

64 3/2M f A1/2H . (18)

One now evaluates f A1/2 on the apparent horizon which in this case will have vanishing

mean curvature, p = 0 to obtain the desired inequality.Actually the term in f involving the cosmological constant was first introduced in [6] and

the results of [5] may need extending to cover this case. I shall assume in what follows that is

possible.

4. Charged black holes

If the energy density takes a particular form then stronger inequalities may be obtained. Thus

in EinsteinMaxwell theory the model metric is the ReissnerNordstromde Sitter one for

which

= k 2Mr

3

r2 +Z2

r 2, (19)


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where

Z2 = Q2 + P2, (20)and P and Q are the electric and magnetic charges. One expects that

2M1

a2

A

4

3/2+ k

A

4

1/2+

4 Z2

A, (21)

where A is the area of the outermost apparent horizon. To obtain this one follows Jang [3] and

notes that

T00 =1

8(E2 +B2), (22)

where E and B are the electric and magnetic fields. They are divergence free with respect to

the metric gij on :

E= 0 = B. (23)The electric charge Q inside a level set is given by

Q = 14

Ss

En dA (24)

where En is the normal component of the electric field E and there is a similar expression for

the magnetic charge P.

Now

T00 1

8(En

2 + Bn2), (25)

and use of the Schwartz inequality givesSs

16 T00 322 Z

2

A(s). (26)

One now integrates the inequality to obtain the desired result.

5. Entropy and attractors

In this section we shall extend the argument above to the case of gravity coupled to scalars

and vectors. In this way we shall vindicate the claim made in [8] that the mass of a black hole

with given charges is never less than the double-extreme hole with the same charges. For more

details about attractors, double-extreme holes, etc and references to the earlier literature the

reader is directed to [8, 9].

Consider a theory with matter Lagrangian

L

= 1

2

()2

1

16

e2 F2. (27)

In general, the scalar will take a particular limiting value at infinity and will vary over theinterior. For static extreme holes it must take a particular value, called in the following frozen

on the horizon. Static extreme black holes for which takes this value everywhere are said

to be double extreme and the scalar is said to be frozen. They have the same geometry as the

extreme ReissnerNordstrom black hole. The mass of any regular static black hole with give

charges (Q,P) is never smaller than the value it takes |Z(q,p)| = |(2|QP|)| for the double-extreme hole with those charges. Since the area of the horizon of a double-extreme hole is


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1682 G W Gibbons

A = 2 |P Q|. One expects that this should be a general lower bound for any time-symmetricinitial data. In fact, we shall show that

2M1

a2

A

4

3/2

+ kA

4

1/2

+4 Z2

A

, (28)

where A is the area of the outermost apparent horizon.

A simple calculation reveals that in the time-symmetric case

T00 =1

2()2 + 1

8e2 (E2 +B2). (29)

However, now

B = 0 = D (30)where the electric induction D is given by

D = e2E. (31)Moreover, now

Q = 14

Ss

Dn dA (32)

where Dn is thenormal componentof theelectricinductionD. Theexpression for themagnetic

charge remains unchanged.

An application of the Schwartz inequality now givesSs

16 T00 21

A2(s)

Ss

(Q2e2 + P2e2 ) dA. (33)

Let us define

V ( , Q , P ) = (Q2e2 + P + 2e2 ). (34)At fixed charges (Q,P), both assumed to be non-vanishing, the function V ( , Q , P ) attains

its least value Z

2

(Q,P) = 2|P Q| at the so-called frozen value = frozen

given byfrozen = 1

2ln |P /Q|. (35)

Thus Ss

16 T00 322 Z

2(Q,P)

A(s). (36)

The result now follows as in the ReissnerNordstrom case.

We have chosen a simple example but it is clear that it extends to include the general class

of theories with any number of Abelian vector and scalar fields coupled to gravity with an

action which is quadratic in the vectors.

6. The second variation

To conclude, we turn to the information about the area of a stable minimal 2-surface in a time-

symmetric slice which comes from the second variation of the area, i.e. the Hessian of the area

functional, and demanding that it be positive. The method is standard. Its application in the

electrovacuum case goes back at least to [17]. For one-parameter variations St = S+ ht ni byan amount h along the normals ni one has

d2A

dt2=

S

h23

Rij ni nj 22

+

S

|h|2 (37)


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where the integral is taken over the minimal surface S and 2 = 12

ab ab is the magnitude

of the trace-free part of the second fundamental form pab of the minimal surface S. The

GaussCodazzi equation gives

2K

G =3 R

23R

ijni nj

22.

(38)Therefore

d2A

dt2=

|h|2 + KGh2

12

3Rh2 2h2

. (39)

If h is taken to be constant over the surface and we use the GaussBonnet theorem then

we must have

4(1 g) A

S

8 T00 > 0, (40)

where the integral is taken over the minimal surface S.

If the cosmological constant is positive and the positive energy condition holds we see

that a stable minimal surface must have spherical topology, an old result. Moreover, its area

must exceed 4/.

If the cosmological constant is positive there will be a cosmological horizon. This is notstable since its area decreases if it is moved uniformly outward or inward. If that is the only

negative mode for the second variation one says that the surface is of index one. Thus the

second variation should be positive ifS

h = 0. (41)

A spectral argument of Yaus [10] gives the following lower bound for a surface of genus g:S

|h|2 + KGh2

S

h2 4(3 g)/A. (42)

Thus ifg = 0 we obtain

AC

12

. (43)This upper bound also follows from Gerochs method. If the inverse mean curvature flow starts

from a black hole horizon of area AH and reaches a cosmological horizon of area AC , one

obtains AH

12

AH

AC

12

AC

, (44)

(see [16] for a recent discussion of related results for a positive cosmological constant).

If the cosmological constant is negative and k = 1 and the positive energy conditionholds, then the second variation yields a lower bound,

A >4(g 1)

|| . (45)

Thus there is a universal topology dependent lowerbound for the entropy of a topologicalblack hole. This resultfits in well with recentideason topological censorship [19] in spacetimes

with a negative cosmological constant. Roughly speaking those authors claim that the topology

of the apparent horizon reflects the topology of infinity. Our lower bound for the area is similar

in that it is not sensitive to detailed properties of the spacetime.

If we are dealing with scalars and vectors, as in the previous section, we know thatS

T00 22 Z

2(Q,P)

A. (46)


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1684 G W Gibbons

This allows a strengthening of the above bounds. Consider the case of black holes with

spherical topology. If the cosmological constant is zero then we have a lower bound for the

entropy with fixed charges, valid for any time-symmetric initial data set:

A > 4Z2. (47)

Again, this substantiates and extends to the non-spherically symmetric case the claims

made in [8].

The results of this and previous sections strongly suggest the idea that

S = Z2 (48)is the irreducible amount of entropy of a black hole with fixed charges (Q,P). This irreducible

amount of entropy is only attained for extreme black holes. Any additional gravitational fields

can only increase it. This is consistent with the microscopic derivations from D-brane theory,

but still leaves open the question of the origin of the extra contribution to the entropy in the

case of non-extreme states. Microscopically this may be due to excited states of D-branes,

but the extrapolation to macroscopic geometric configurations of classical initial data in the

non-BPS situations is not at all obvious.

7. The HorowitzMyers conjecture

One motivation for this present work was to see whether one might use the inverse mean

curvature flow to tackle an interesting conjecture of Horowtiz and Myers [11]. This conjecture

wasmadeinthecontextoftheAdS5/CFTcorrespondencewhich concerns (4+1)-dimensional

spacetimes. However, the mean curvature flow method does not seem to work except in the

case of three-dimensional initial data sets and therefore it is not applicable to the original

HorowitzMyers conjecture. However, there is a version of the HorowitzMyers conjecture

for p-dimensional initial data sets for all p, including the case p = 3 where one might havehoped that the inverse mean-curvature method would have been applicable. Unfortunately it

is not. To see why, we recall that HorowitzMyers example is obtained by interchanging the

role of time with one of the torus coordinates in the k=

0 Kottler metrics:

ds2 = r2

a2dt2 +

dr 2

((r 2/a2) (r 30 /a2r))+

r2

a2 r

30

a2r

dx2 +

r 2

a2dy2. (49)

Regularity demands that we identify x with period = 4 a2/3r0. The period of thecoordinate y, call it Ly is arbitrary. The initial data set thus has topology R

2 S1. Note thatthe spacetime has no event horizon. Horowitz and Myers show that the ADM mass (obtained

by comparing with the case r0 = 0 in a manner that they explain in detail) is negative andequals

42Ly a

2

27L2x. (50)

Consider initial data sets which are asymptotic to the hyperbolic metric obtained by setting

dt

=0 in (49). They give evidence to support the conjecture that this is a lower bound for

the ADM mass of such data sets, specified by Lx and Ly . As in the case of the negativemass k = 1 black holes this suggestion is striking because the lower bound is attained for anon-BPS state.

More precisely, consider complete non-singular symptotically locally hyperbolic 3-

manifolds with scalar curvature 3R = 6a2 which are asymptotic (with corrections O(1/r 2))to the metric

ds2 = dr2

((r 2/a2) (r 30 /a2r))+

r2

a2 r

30

a2r

dx2 +

r 2

a2dy2, (51)


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with 0 x and the period ofy arbitrary (we take it to be infinite). Horowitz and Myers

conjecture that

r0 4 a2

3

, (52)

with equality if and only if the metric is isometric to (51). Note that if equality holds then (51)

is a complete non-singular metric on R2 R, where theR factor corresponds to the coordinatey. The isometry group is S1 R where the circle action corresponding to translation in x hasa a line of fixed points at r = r0.

The difficulty with applying the inverse mean curvature flow is that because the spacetime

has no event horizon, the initial data set need not have, and indeed does not have, an apparent

horizon. Thus there is no natural 2-surface from which to start the flow. The fixed point set at

r = r0 of the circle action in the metric (51) is a geodesic. One might be tempted to assumethat such a geodesic exits in the metric whose mass we are trying to bound. We could try to

set the flow off from that geodesic. The problem is that if we consider a small tube of radius r

surrounding the geodesic, we see that the function f would tend to minus infinity like r1 aswe shrink the tube onto the geodesic. Since the area of the tube shrinks as r the product

Af

diverges like r1/2. The monotonicity property thus seems to give nothing interesting. On theother, had if we were to start the flow from a very small sphere we would, assuming that the

flow reaches infinity, merely obtain the result that the quantity r0 is positive.

8. Cosmic censorship versus Bogomolnyi

If < 0 and k = 1 we obtain the cosmic censorship lower bound

M rH +r2H

a2+

Z2

rH. (53)

By extremizing with respect to rH we find, after some elementary algebra and manipulation

of surds, the following interesting the lower bound for M in terms of|Z| and

M

2

3|Z|

1 +

1 4Z2 + 1

1 +

1 4Z2

. (54)

Of course in the limit that 0 we recover the familiar Bogomolnyi bound:M |Z|, (55)

but it is clear that the cosmic censorship lower bound (54) is strictly greater than the

Bogomolnyi lower bound (55) if < 0.

9. Higher dimensions

As originally formulated the Geroch technique works only for three spatial dimensional initial

data sets. However, there is an analogue of the positive mass theorem in all higher dimensions.This has been proved by Schoen and Yau and by Witten. A reformulation of Gerochs idea

designed to see how this might work in higher dimensions (shown to me by Douglas Eardley

in 1980) goes as follows: one considers a foliation of an n-dimensional manifold of the form

ds2 = e2 dr2 + r2hab (x,r) dxa dxb (56)where

rdet hab = 0. (57)


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1686 G W Gibbons

Thus the trace of the second fundamental form is

p = n 1r

. (58)

Let

qab =1

2e

rhab (59)

so that

qab =1

r 2

pab

1

n 1 hab p

. (60)

The Ricci scalar is given by

nR = 1r n1

r

(n 1)r n2(1 e2 )

2

r 2e DaDae qab qab

+1

r 2

n1R (n 1)(n 2)

(61)

where n

1R is the scalar curvature of the level sets r=

constant.

One may now multiply this equation by rn1det hab dr dn1x and integrate to obtainthe relevant identities. One assumes that the outermost apparent horizon is located at r = rHon which = 0. If the metric is asymptotically flat, it is convenient to normalize such that

Sr

det hab d

n1x = vol(Sn1), (62)

where vol(Sn1) is the volume of a standard round sphere of unit radius. Then at large distances

a2rn2

(63)

where the total mass M is given by

M

=

(n 1)vol(Sn1)

16

a. (64)

We obtain the integral inequality

a r n2H +

F rn3 dr (65)

where

F(r) =

Sr

det hab d

n1x

n1R (n 2)(n 1)

. (66)

To make the theorems work we need this to be non-positive. Ifn = 3 this follows from theGaussBonnet theorem. For n > 3 the situation is less clear. The integrand vanishes for a

round sphere and this fact leads to some obvious positivity properties if we assume that the

levels sets are the orbits of an SO(n) isometry group. If the metric is not SO(n)-invariant

however we have, without further information, little control of the integrand. The quantity F

is, of course, related to the Yamabe constant. The Yamabe constant is a conformal invariant

of a closed d-dimensional Riemannian manifold {M, g} and is given, up to a factor whichdepends on conventions, by the infinum over all metrics g in the same conformal equivalenceclass as the metric g of

M

R/(Vol(M))(d2)/d, (67)

where R is the scalar curvature of the metric g.


12/12


The fact that this constant has already arisen in the context of entropy in cosmology see

[14, 15, 20] is striking.

Note that to derive inequalities from the second variation (40) we needed to use the Gauss

Bonnet theorem. In higher dimensions what appears in the second variation is the quantityS

R (68)

where R is the Ricci scalar of the apparent horizon. If the cosmological constant is non-

negative this must be positive. This would place some topological restrictions on the apparent

horizon.

Thus it seems that the only real evidence for a bound to the area of apparent horizons

comes from the higher-dimensional versions of the collapsing shell calculations [18].

Acknowledgments

I should like to thank Gary Horowitz and Tom Ilmanen for a number of helpful discussions.

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